NCERT Mathematics Class 12 - Chapter 1: Relations and Functions - Notes

Learning Objectives

• Understand types of relations: reflexive, symmetric, transitive, equivalence
• Learn types of functions: one-one, onto, bijective
• Study composition of functions and inverse functions
• Understand binary operations

Key Concepts

Types of Relations

A relation R on set A is a subset of A × A. Reflexive: (a, a) ∈ R for all a ∈ A. Symmetric: (a, b) ∈ R implies (b, a) ∈ R. Transitive: (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R. Equivalence relation: Reflexive + Symmetric + Transitive. An equivalence relation partitions a set into disjoint equivalence classes.

Empty relation: R = φ (no element is related). Universal relation: R = A × A (every element is related to every element). Universal relation is always an equivalence relation. Empty relation is symmetric and transitive but not reflexive (unless A = φ).

Types of Functions

One-one (Injective): f(a) = f(b) ⟹ a = b. To check: show that f(x₁) = f(x₂) implies x₁ = x₂, or show f is strictly monotonic (increasing or decreasing). Onto (Surjective): For every y ∈ codomain, there exists x ∈ domain such that f(x) = y (range = codomain). Bijective: Both one-one and onto. A bijective function has an inverse.

Number of bijections from A to A (|A| = n) = n!. A function from finite set to itself is one-one iff it is onto.

Composition of Functions

If f: A → B and g: B → C, then gof: A → C defined by (gof)(x) = g(f(x)). Composition is not commutative: gof ≠ fog in general. Composition is associative: ho(gof) = (hog)of. If f and g are both one-one, then gof is one-one. If f and g are both onto, then gof is onto.

Inverse of a Function

If f: A → B is bijective, then f^-1: B → A exists such that f^-1(y) = x iff f(x) = y. Properties: fof^-1 = I_B (identity on B); f^-1of = I_A (identity on A). (gof)^-1 = f^-1og^-1.

Binary Operations

A binary operation * on set A is a function *: A × A → A. Commutative: a * b = b * a. Associative: (a * b) * c = a * (b * c). Identity element e: a * e = e * a = a for all a. Inverse of a: a * b = b * a = e. Example: Addition is commutative, associative on Z with identity 0.

Summary

Relations can be reflexive, symmetric, transitive, or equivalence relations. Functions are classified as injective, surjective, or bijective. Only bijective functions have inverses. Composition of functions is associative but not commutative. Binary operations on sets follow algebraic properties.

Important Terms

• Equivalence relation: Reflexive + Symmetric + Transitive
• Bijective function: One-one and onto; has inverse
• Composition: (gof)(x) = g(f(x)); not commutative
• Inverse function: f^-1 exists only when f is bijective
• Identity element: Element e such that a * e = a for all a
• Equivalence class: Set of elements related to each other

Quick Revision

• Equivalence relation = reflexive + symmetric + transitive
• One-one: f(a)=f(b) ⟹ a=b; Onto: range = codomain
• Bijective function has unique inverse
• gof ≠ fog; but ho(gof) = (hog)of
• (gof)^-1 = f^-1og^-1
• Number of bijections from set of n elements to itself = n!
• Binary operation must be closed (result in same set)